Dirichlet series
| L(s) = 1 | − 12·2-s − 12·3-s + 78·4-s − 3·5-s + 144·6-s − 7-s − 364·8-s + 78·9-s + 36·10-s − 7·11-s − 936·12-s + 4·13-s + 12·14-s + 36·15-s + 1.36e3·16-s − 4·17-s − 936·18-s − 12·19-s − 234·20-s + 12·21-s + 84·22-s − 18·23-s + 4.36e3·24-s − 15·25-s − 48·26-s − 364·27-s − 78·28-s + ⋯ |
| L(s) = 1 | − 8.48·2-s − 6.92·3-s + 39·4-s − 1.34·5-s + 58.7·6-s − 0.377·7-s − 128.·8-s + 26·9-s + 11.3·10-s − 2.11·11-s − 270.·12-s + 1.10·13-s + 3.20·14-s + 9.29·15-s + 341.·16-s − 0.970·17-s − 220.·18-s − 2.75·19-s − 52.3·20-s + 2.61·21-s + 17.9·22-s − 3.75·23-s + 891.·24-s − 3·25-s − 9.41·26-s − 70.0·27-s − 14.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.59035\times 10^{20}\) |
| Root analytic conductor: | \(6.94590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{12} \) |
| 3 | \( ( 1 + T )^{12} \) | |
| 19 | \( ( 1 + T )^{12} \) | |
| 53 | \( ( 1 + T )^{12} \) | |
| good | 5 | \( 1 + 3 T + 24 T^{2} + 12 p T^{3} + 63 p T^{4} + 687 T^{5} + 2932 T^{6} + 5977 T^{7} + 879 p^{2} T^{8} + 41708 T^{9} + 137214 T^{10} + 243173 T^{11} + 732886 T^{12} + 243173 p T^{13} + 137214 p^{2} T^{14} + 41708 p^{3} T^{15} + 879 p^{6} T^{16} + 5977 p^{5} T^{17} + 2932 p^{6} T^{18} + 687 p^{7} T^{19} + 63 p^{9} T^{20} + 12 p^{10} T^{21} + 24 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + T + 24 T^{2} + 37 T^{3} + 340 T^{4} + 613 T^{5} + 564 p T^{6} + 6444 T^{7} + 39516 T^{8} + 58027 T^{9} + 327844 T^{10} + 483768 T^{11} + 2379822 T^{12} + 483768 p T^{13} + 327844 p^{2} T^{14} + 58027 p^{3} T^{15} + 39516 p^{4} T^{16} + 6444 p^{5} T^{17} + 564 p^{7} T^{18} + 613 p^{7} T^{19} + 340 p^{8} T^{20} + 37 p^{9} T^{21} + 24 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 7 T + 68 T^{2} + 402 T^{3} + 2494 T^{4} + 12378 T^{5} + 61262 T^{6} + 268473 T^{7} + 1140523 T^{8} + 4506422 T^{9} + 16999038 T^{10} + 60867808 T^{11} + 206716044 T^{12} + 60867808 p T^{13} + 16999038 p^{2} T^{14} + 4506422 p^{3} T^{15} + 1140523 p^{4} T^{16} + 268473 p^{5} T^{17} + 61262 p^{6} T^{18} + 12378 p^{7} T^{19} + 2494 p^{8} T^{20} + 402 p^{9} T^{21} + 68 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 4 T + 64 T^{2} - 190 T^{3} + 2096 T^{4} - 4714 T^{5} + 47766 T^{6} - 86875 T^{7} + 900769 T^{8} - 1467767 T^{9} + 1144966 p T^{10} - 22766494 T^{11} + 211030308 T^{12} - 22766494 p T^{13} + 1144966 p^{3} T^{14} - 1467767 p^{3} T^{15} + 900769 p^{4} T^{16} - 86875 p^{5} T^{17} + 47766 p^{6} T^{18} - 4714 p^{7} T^{19} + 2096 p^{8} T^{20} - 190 p^{9} T^{21} + 64 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T + 126 T^{2} + 478 T^{3} + 7928 T^{4} + 29268 T^{5} + 332220 T^{6} + 1182395 T^{7} + 10277153 T^{8} + 34538639 T^{9} + 246444654 T^{10} + 762843696 T^{11} + 4688561020 T^{12} + 762843696 p T^{13} + 246444654 p^{2} T^{14} + 34538639 p^{3} T^{15} + 10277153 p^{4} T^{16} + 1182395 p^{5} T^{17} + 332220 p^{6} T^{18} + 29268 p^{7} T^{19} + 7928 p^{8} T^{20} + 478 p^{9} T^{21} + 126 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 18 T + 277 T^{2} + 2849 T^{3} + 26965 T^{4} + 8969 p T^{5} + 1504353 T^{6} + 9486742 T^{7} + 58464451 T^{8} + 322192772 T^{9} + 1766231674 T^{10} + 8812006758 T^{11} + 44173025294 T^{12} + 8812006758 p T^{13} + 1766231674 p^{2} T^{14} + 322192772 p^{3} T^{15} + 58464451 p^{4} T^{16} + 9486742 p^{5} T^{17} + 1504353 p^{6} T^{18} + 8969 p^{8} T^{19} + 26965 p^{8} T^{20} + 2849 p^{9} T^{21} + 277 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 10 T + 201 T^{2} - 1510 T^{3} + 17045 T^{4} - 103782 T^{5} + 859615 T^{6} - 4372612 T^{7} + 29965380 T^{8} - 128207764 T^{9} + 820476684 T^{10} - 3150875206 T^{11} + 22069839308 T^{12} - 3150875206 p T^{13} + 820476684 p^{2} T^{14} - 128207764 p^{3} T^{15} + 29965380 p^{4} T^{16} - 4372612 p^{5} T^{17} + 859615 p^{6} T^{18} - 103782 p^{7} T^{19} + 17045 p^{8} T^{20} - 1510 p^{9} T^{21} + 201 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 14 T + 262 T^{2} - 2649 T^{3} + 31175 T^{4} - 264176 T^{5} + 2441356 T^{6} - 18016911 T^{7} + 139806331 T^{8} - 912412221 T^{9} + 6160616494 T^{10} - 35877413715 T^{11} + 214389199162 T^{12} - 35877413715 p T^{13} + 6160616494 p^{2} T^{14} - 912412221 p^{3} T^{15} + 139806331 p^{4} T^{16} - 18016911 p^{5} T^{17} + 2441356 p^{6} T^{18} - 264176 p^{7} T^{19} + 31175 p^{8} T^{20} - 2649 p^{9} T^{21} + 262 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 14 T + 340 T^{2} - 3986 T^{3} + 56288 T^{4} - 559922 T^{5} + 5935740 T^{6} - 50824307 T^{7} + 442524299 T^{8} - 89074589 p T^{9} + 24531661484 T^{10} - 159907974214 T^{11} + 1036081982592 T^{12} - 159907974214 p T^{13} + 24531661484 p^{2} T^{14} - 89074589 p^{4} T^{15} + 442524299 p^{4} T^{16} - 50824307 p^{5} T^{17} + 5935740 p^{6} T^{18} - 559922 p^{7} T^{19} + 56288 p^{8} T^{20} - 3986 p^{9} T^{21} + 340 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 9 T + 206 T^{2} + 1144 T^{3} + 18788 T^{4} + 66254 T^{5} + 1119533 T^{6} + 1821259 T^{7} + 52493749 T^{8} + 5197468 T^{9} + 56098733 p T^{10} - 1680157150 T^{11} + 96255910220 T^{12} - 1680157150 p T^{13} + 56098733 p^{3} T^{14} + 5197468 p^{3} T^{15} + 52493749 p^{4} T^{16} + 1821259 p^{5} T^{17} + 1119533 p^{6} T^{18} + 66254 p^{7} T^{19} + 18788 p^{8} T^{20} + 1144 p^{9} T^{21} + 206 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 4 T + 153 T^{2} + 151 T^{3} + 11075 T^{4} - 8019 T^{5} + 696245 T^{6} - 573064 T^{7} + 42683803 T^{8} - 26391044 T^{9} + 2192892010 T^{10} - 2234107318 T^{11} + 96420458114 T^{12} - 2234107318 p T^{13} + 2192892010 p^{2} T^{14} - 26391044 p^{3} T^{15} + 42683803 p^{4} T^{16} - 573064 p^{5} T^{17} + 696245 p^{6} T^{18} - 8019 p^{7} T^{19} + 11075 p^{8} T^{20} + 151 p^{9} T^{21} + 153 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 14 T + 397 T^{2} + 4457 T^{3} + 71645 T^{4} + 670595 T^{5} + 168698 p T^{6} + 63746013 T^{7} + 617137951 T^{8} + 4390698577 T^{9} + 37237575967 T^{10} + 241901661008 T^{11} + 1881552850466 T^{12} + 241901661008 p T^{13} + 37237575967 p^{2} T^{14} + 4390698577 p^{3} T^{15} + 617137951 p^{4} T^{16} + 63746013 p^{5} T^{17} + 168698 p^{7} T^{18} + 670595 p^{7} T^{19} + 71645 p^{8} T^{20} + 4457 p^{9} T^{21} + 397 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 18 T + 663 T^{2} + 9600 T^{3} + 200429 T^{4} + 2424716 T^{5} + 36988255 T^{6} + 382245750 T^{7} + 4673888770 T^{8} + 41762236780 T^{9} + 427749299366 T^{10} + 3319410455900 T^{11} + 29134332721224 T^{12} + 3319410455900 p T^{13} + 427749299366 p^{2} T^{14} + 41762236780 p^{3} T^{15} + 4673888770 p^{4} T^{16} + 382245750 p^{5} T^{17} + 36988255 p^{6} T^{18} + 2424716 p^{7} T^{19} + 200429 p^{8} T^{20} + 9600 p^{9} T^{21} + 663 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 7 T + 324 T^{2} + 1973 T^{3} + 52579 T^{4} + 273984 T^{5} + 5885543 T^{6} + 26544167 T^{7} + 519606493 T^{8} + 34330308 p T^{9} + 38811764973 T^{10} + 143665347805 T^{11} + 2528142756094 T^{12} + 143665347805 p T^{13} + 38811764973 p^{2} T^{14} + 34330308 p^{4} T^{15} + 519606493 p^{4} T^{16} + 26544167 p^{5} T^{17} + 5885543 p^{6} T^{18} + 273984 p^{7} T^{19} + 52579 p^{8} T^{20} + 1973 p^{9} T^{21} + 324 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 8 T + 489 T^{2} - 4391 T^{3} + 124115 T^{4} - 1136315 T^{5} + 21277365 T^{6} - 186397314 T^{7} + 2695009947 T^{8} - 21722177986 T^{9} + 261837198506 T^{10} - 1897963105728 T^{11} + 19826358261602 T^{12} - 1897963105728 p T^{13} + 261837198506 p^{2} T^{14} - 21722177986 p^{3} T^{15} + 2695009947 p^{4} T^{16} - 186397314 p^{5} T^{17} + 21277365 p^{6} T^{18} - 1136315 p^{7} T^{19} + 124115 p^{8} T^{20} - 4391 p^{9} T^{21} + 489 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + T + 517 T^{2} + 1289 T^{3} + 130947 T^{4} + 438001 T^{5} + 22169645 T^{6} + 79615669 T^{7} + 2800760467 T^{8} + 9815291020 T^{9} + 275988974622 T^{10} + 906340894864 T^{11} + 21776891780370 T^{12} + 906340894864 p T^{13} + 275988974622 p^{2} T^{14} + 9815291020 p^{3} T^{15} + 2800760467 p^{4} T^{16} + 79615669 p^{5} T^{17} + 22169645 p^{6} T^{18} + 438001 p^{7} T^{19} + 130947 p^{8} T^{20} + 1289 p^{9} T^{21} + 517 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 31 T + 834 T^{2} + 16630 T^{3} + 296136 T^{4} + 4532874 T^{5} + 63727178 T^{6} + 807398767 T^{7} + 9514617687 T^{8} + 103108757858 T^{9} + 1047266275604 T^{10} + 9867616205064 T^{11} + 87411789698432 T^{12} + 9867616205064 p T^{13} + 1047266275604 p^{2} T^{14} + 103108757858 p^{3} T^{15} + 9514617687 p^{4} T^{16} + 807398767 p^{5} T^{17} + 63727178 p^{6} T^{18} + 4532874 p^{7} T^{19} + 296136 p^{8} T^{20} + 16630 p^{9} T^{21} + 834 p^{10} T^{22} + 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 358 T^{2} - 1145 T^{3} + 79396 T^{4} - 4280 p T^{5} + 13001361 T^{6} - 64727295 T^{7} + 1670112273 T^{8} - 8441409187 T^{9} + 176988554865 T^{10} - 852293310995 T^{11} + 15293019262212 T^{12} - 852293310995 p T^{13} + 176988554865 p^{2} T^{14} - 8441409187 p^{3} T^{15} + 1670112273 p^{4} T^{16} - 64727295 p^{5} T^{17} + 13001361 p^{6} T^{18} - 4280 p^{8} T^{19} + 79396 p^{8} T^{20} - 1145 p^{9} T^{21} + 358 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 + 48 T + 1548 T^{2} + 36442 T^{3} + 709298 T^{4} + 11656196 T^{5} + 168584118 T^{6} + 2168483725 T^{7} + 25427798483 T^{8} + 274419632687 T^{9} + 2784745224590 T^{10} + 26813220647848 T^{11} + 248918594206548 T^{12} + 26813220647848 p T^{13} + 2784745224590 p^{2} T^{14} + 274419632687 p^{3} T^{15} + 25427798483 p^{4} T^{16} + 2168483725 p^{5} T^{17} + 168584118 p^{6} T^{18} + 11656196 p^{7} T^{19} + 709298 p^{8} T^{20} + 36442 p^{9} T^{21} + 1548 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 13 T + 649 T^{2} - 7625 T^{3} + 216783 T^{4} - 2304145 T^{5} + 48195601 T^{6} - 463705509 T^{7} + 7882775667 T^{8} - 68571962902 T^{9} + 995382856534 T^{10} - 7793439760542 T^{11} + 99313353481338 T^{12} - 7793439760542 p T^{13} + 995382856534 p^{2} T^{14} - 68571962902 p^{3} T^{15} + 7882775667 p^{4} T^{16} - 463705509 p^{5} T^{17} + 48195601 p^{6} T^{18} - 2304145 p^{7} T^{19} + 216783 p^{8} T^{20} - 7625 p^{9} T^{21} + 649 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 25 T + 1092 T^{2} - 22092 T^{3} + 552554 T^{4} - 9264404 T^{5} + 171023956 T^{6} - 2425271421 T^{7} + 36117670607 T^{8} - 439125683202 T^{9} + 5491951115368 T^{10} - 57636033718824 T^{11} + 617208445099148 T^{12} - 57636033718824 p T^{13} + 5491951115368 p^{2} T^{14} - 439125683202 p^{3} T^{15} + 36117670607 p^{4} T^{16} - 2425271421 p^{5} T^{17} + 171023956 p^{6} T^{18} - 9264404 p^{7} T^{19} + 552554 p^{8} T^{20} - 22092 p^{9} T^{21} + 1092 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.73461669617673815882263532283, −2.39801207024205425817159112332, −2.37188651161314828423694866238, −2.28366041020708827333225845640, −2.26605108027438659842603369028, −2.23740556503985550547882174533, −2.12292192997046325534049130455, −2.11491558832029915108065486155, −2.03875252833173399616488369489, −2.02698585620351945354424181219, −1.92859731773454682710742275901, −1.91986902625605130382970634790, −1.90225576460213147792789596452, −1.36629981898347530261697472056, −1.34574726537964986671072293595, −1.31954205074999770848494450594, −1.26830118886571971243176902609, −1.22232141594391692347187308266, −1.21106111255796264483080429688, −1.18413226881823111211370128623, −1.15297517597175311343071819847, −1.12905615624022191052027749223, −0.899670331681225878305189241440, −0.77803566384066747937630543196, −0.75782308489676759133616488992, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.75782308489676759133616488992, 0.77803566384066747937630543196, 0.899670331681225878305189241440, 1.12905615624022191052027749223, 1.15297517597175311343071819847, 1.18413226881823111211370128623, 1.21106111255796264483080429688, 1.22232141594391692347187308266, 1.26830118886571971243176902609, 1.31954205074999770848494450594, 1.34574726537964986671072293595, 1.36629981898347530261697472056, 1.90225576460213147792789596452, 1.91986902625605130382970634790, 1.92859731773454682710742275901, 2.02698585620351945354424181219, 2.03875252833173399616488369489, 2.11491558832029915108065486155, 2.12292192997046325534049130455, 2.23740556503985550547882174533, 2.26605108027438659842603369028, 2.28366041020708827333225845640, 2.37188651161314828423694866238, 2.39801207024205425817159112332, 2.73461669617673815882263532283
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.